A theorem of Giraud says that gerbes over a scheme $X$ bounded by a sheaf of Abelian groups $A$ are classified by elements of the etale cohomology group $H^2(X,A)$. Similar stateme …

If $G$ is a profinite group, then the category $Mod(G)$ of discrete $G$-modules has sufficiently many injectives (Neukirch, Schmidt, Wingberg: Cohomology of Number Fields, 2.6.5). …

Let $X$ be an algebraic variety over an algebraically closed field $k$ and let a finite group $G$ act on it so that it acts freely on the generic fibre of the projection $X \to X/G …

I've recently been reading about Euler systems, and was curious where the name comes from. In particular, while the notion of an Euler system is still not rigorously defined, does …

Let $X$ a smooth projective geometrically connected curve over a finite field $k$. Let $J$ a smooth commutative group scheme over $X$ and $F$ the function field of $X$. Do we have …

For writing a local class field theory using Galois cohomology, maybe first step is to determine a Braue group of a local field. it is known that Brauer group of a local field is i …

I have been working through Langlands paper which you can see here http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/AbelianAlg-ps.pdf and I can understand why one of his …

Let $C$ be a (quasi-projective) variety over an algebraically closed field $k$ and let $k(C)$ be its field of rational functions. Then for any birational map $\sigma: C \dashrighta …

Regarding Gregory Berhuy's book "An Introduction to Galois Cohomology and its Applications": The book defined a cohomology sets for non-abelian $G$-groups. Let $A$ be a $G$-group, …

Let $X$ be a smooth projective geometrically connected curve over a number field $k$ of genus $g\geq 2$. Is the number of twists of $X$ always infinite? (The answer is no, because …

Let $k$ be a local field of char $0$ (which is the case I concern). Let $V$ be a variety defined over $k$ and let $f: \mathbb A^n\to V$ be a surjective map (over the algebraic c …

Among the many equivalent formulations of Leopoldt's conjecture, this one is probably the shortest: For any number field $K$, prime number $p$, finite set $S$ of primes of $K$ cont …

The automorphism group of the algebra of $n$-dimensional matrices over a field $K$ is $PGL_n(K)$. The automorphism group of $n-1$-dimensional projective space over $K$ is also $PGL …

For Tate twists Z_p(2), which is defined by the projective limit of \mu_{p^m}(2) over all m>0, I would like to calculate H^1(Q_p, Z_p(2)). I guess this is zero, but cannot prove …

Suppose $K$ is a finite extension of $\mathbb{Q}_p$ and $K^{nr}$ the maximal unramified extension of $K$ in some fixed algebraic closure. Let $G_K$ be the absolute Galois group of …